Vector Length and Angle Practice Worksheet

Find the length of the vector v = (2, 3, 6).

Square the components: 4, 9, 36. Add them to get 49. Take the square root: the length is 7.

Find the length of the vector u = (-1, 2, -2).

Square the components: 1, 4, 4. Add them to get 9. Take the square root: the length is 3.

Calculate the scalar product of a = (3, -1, 4) and b = (2, 5, 1).

Multiply matching components: (3)(2) + (-1)(5) + (4)(1) = 6 - 5 + 4 = 5.

Determine whether vectors u = (1, 2, 3) and v = (2, 1, -4) are perpendicular.

Compute the dot product: (1)(2) + (2)(1) + (3)(-4) = -8. Because the scalar product is not zero, they are not perpendicular. The negative sign shows the angle is obtuse.

Find the angle between the vectors p = (1, 0, 0) and q = (0, 1, 0).

The dot product is (1)(0) + (0)(1) + (0)(0) = 0. The lengths are both 1. Cosine theta = 0 / 1 = 0, so the angle is 90 degrees.

Find the exact angle between u = (1, 1, 0) and v = (0, 1, 1).

Dot product: 0 + 1 + 0 = 1. Length of u is sqrt(2), length of v is sqrt(2). Product of lengths is 2. Cosine theta = 1/2, so the angle is 60 degrees.